4,027 research outputs found
On variable-weighted exact satisfiability problems
We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(|C|2^{0.2441n}) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n^2cdot |C|+2^{0.40567n}) time. In recent years only the unweighted counterparts of these problems have been studied cite{dahl,dahl2,porschen}
On variable-weighted exact satisfiability problems
We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(|C|2^{0.2441n}) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n^2cdot |C|+2^{0.40567n}) time. In recent years only the unweighted counterparts of these problems have been studied cite{dahl,dahl2,porschen}
Fuzzy Maximum Satisfiability
In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to
{\L}ukasiewicz logic. The MaxSAT problem for a set of formulae {\Phi} is the
problem of finding an assignment to the variables in {\Phi} that satisfies the
maximum number of formulae. Three possible solutions (encodings) are proposed
to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer
Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem
(WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have
numerous applications in optimization problems that involve vagueness.Comment: 10 page
A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits
We give a nontrivial algorithm for the satisfiability problem for cn-wire
threshold circuits of depth two which is better than exhaustive search by a
factor 2^{sn} where s= 1/c^{O(c^2)}. We believe that this is the first
nontrivial satisfiability algorithm for cn-wire threshold circuits of depth
two. The independently interesting problem of the feasibility of sparse 0-1
integer linear programs is a special case. To our knowledge, our algorithm is
the first to achieve constant savings even for the special case of Integer
Linear Programming. The key idea is to reduce the satisfiability problem to the
Vector Domination Problem, the problem of checking whether there are two
vectors in a given collection of vectors such that one dominates the other
component-wise.
We also provide a satisfiability algorithm with constant savings for depth
two circuits with symmetric gates where the total weighted fan-in is at most
cn.
One of our motivations is proving strong lower bounds for TC^0 circuits,
exploiting the connection (established by Williams) between satisfiability
algorithms and lower bounds. Our second motivation is to explore the connection
between the expressive power of the circuits and the complexity of the
corresponding circuit satisfiability problem
Probabilistic Inference Modulo Theories
We present SGDPLL(T), an algorithm that solves (among many other problems)
probabilistic inference modulo theories, that is, inference problems over
probabilistic models defined via a logic theory provided as a parameter
(currently, propositional, equalities on discrete sorts, and inequalities, more
specifically difference arithmetic, on bounded integers). While many solutions
to probabilistic inference over logic representations have been proposed,
SGDPLL(T) is simultaneously (1) lifted, (2) exact and (3) modulo theories, that
is, parameterized by a background logic theory. This offers a foundation for
extending it to rich logic languages such as data structures and relational
data. By lifted, we mean algorithms with constant complexity in the domain size
(the number of values that variables can take). We also detail a solver for
summations with difference arithmetic and show experimental results from a
scenario in which SGDPLL(T) is much faster than a state-of-the-art
probabilistic solver.Comment: Submitted to StarAI-16 workshop as closely revised version of
IJCAI-16 pape
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